let S1, S2, S be non empty non void Circuit-like ManySortedSign ; :: thesis: ( InputVertices S1 misses InnerVertices S2 & InputVertices S2 misses InnerVertices S1 & S = S1 +* S2 implies for A1 being non-empty Circuit of S1

for A2 being non-empty Circuit of S2

for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds

for s being State of A

for s1 being State of A1 st s1 = s | the carrier of S1 holds

for s2 being State of A2 st s2 = s | the carrier of S2 holds

for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n)) )

assume that

A1: InputVertices S1 misses InnerVertices S2 and

A2: InputVertices S2 misses InnerVertices S1 and

A3: S = S1 +* S2 ; :: thesis: for A1 being non-empty Circuit of S1

for A2 being non-empty Circuit of S2

for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds

for s being State of A

for s1 being State of A1 st s1 = s | the carrier of S1 holds

for s2 being State of A2 st s2 = s | the carrier of S2 holds

for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n))

let A1 be non-empty Circuit of S1; :: thesis: for A2 being non-empty Circuit of S2

for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds

for s being State of A

for s1 being State of A1 st s1 = s | the carrier of S1 holds

for s2 being State of A2 st s2 = s | the carrier of S2 holds

for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n))

let A2 be non-empty Circuit of S2; :: thesis: for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds

for s being State of A

for s1 being State of A1 st s1 = s | the carrier of S1 holds

for s2 being State of A2 st s2 = s | the carrier of S2 holds

for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n))

let A be non-empty Circuit of S; :: thesis: ( A1 tolerates A2 & A = A1 +* A2 implies for s being State of A

for s1 being State of A1 st s1 = s | the carrier of S1 holds

for s2 being State of A2 st s2 = s | the carrier of S2 holds

for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n)) )

assume that

A4: A1 tolerates A2 and

A5: A = A1 +* A2 ; :: thesis: for s being State of A

for s1 being State of A1 st s1 = s | the carrier of S1 holds

for s2 being State of A2 st s2 = s | the carrier of S2 holds

for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n))

let s be State of A; :: thesis: for s1 being State of A1 st s1 = s | the carrier of S1 holds

for s2 being State of A2 st s2 = s | the carrier of S2 holds

for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n))

let s1 be State of A1; :: thesis: ( s1 = s | the carrier of S1 implies for s2 being State of A2 st s2 = s | the carrier of S2 holds

for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n)) )

assume A6: s1 = s | the carrier of S1 ; :: thesis: for s2 being State of A2 st s2 = s | the carrier of S2 holds

for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n))

let s2 be State of A2; :: thesis: ( s2 = s | the carrier of S2 implies for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n)) )

assume A7: s2 = s | the carrier of S2 ; :: thesis: for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n))

let n be natural Number ; :: thesis: Following (s,n) = (Following (s1,n)) +* (Following (s2,n))

A8: (Following (s,n)) | the carrier of S1 = Following (s1,n) by A1, A3, A4, A5, A6, Th13;

A9: ( dom (Following (s,n)) = the carrier of S & the carrier of S = the carrier of S1 \/ the carrier of S2 ) by A3, CIRCCOMB:def 2, CIRCUIT1:3;

S1 tolerates S2 by A4, CIRCCOMB:def 3;

then A10: S1 +* S2 = S2 +* S1 by CIRCCOMB:5;

A1 +* A2 = A2 +* A1 by A4, CIRCCOMB:22;

then (Following (s,n)) | the carrier of S2 = Following (s2,n) by A2, A3, A4, A5, A7, A10, Th13, CIRCCOMB:19;

hence Following (s,n) = (Following (s1,n)) +* (Following (s2,n)) by A8, A9, FUNCT_4:70; :: thesis: verum

for A2 being non-empty Circuit of S2

for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds

for s being State of A

for s1 being State of A1 st s1 = s | the carrier of S1 holds

for s2 being State of A2 st s2 = s | the carrier of S2 holds

for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n)) )

assume that

A1: InputVertices S1 misses InnerVertices S2 and

A2: InputVertices S2 misses InnerVertices S1 and

A3: S = S1 +* S2 ; :: thesis: for A1 being non-empty Circuit of S1

for A2 being non-empty Circuit of S2

for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds

for s being State of A

for s1 being State of A1 st s1 = s | the carrier of S1 holds

for s2 being State of A2 st s2 = s | the carrier of S2 holds

for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n))

let A1 be non-empty Circuit of S1; :: thesis: for A2 being non-empty Circuit of S2

for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds

for s being State of A

for s1 being State of A1 st s1 = s | the carrier of S1 holds

for s2 being State of A2 st s2 = s | the carrier of S2 holds

for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n))

let A2 be non-empty Circuit of S2; :: thesis: for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds

for s being State of A

for s1 being State of A1 st s1 = s | the carrier of S1 holds

for s2 being State of A2 st s2 = s | the carrier of S2 holds

for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n))

let A be non-empty Circuit of S; :: thesis: ( A1 tolerates A2 & A = A1 +* A2 implies for s being State of A

for s1 being State of A1 st s1 = s | the carrier of S1 holds

for s2 being State of A2 st s2 = s | the carrier of S2 holds

for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n)) )

assume that

A4: A1 tolerates A2 and

A5: A = A1 +* A2 ; :: thesis: for s being State of A

for s1 being State of A1 st s1 = s | the carrier of S1 holds

for s2 being State of A2 st s2 = s | the carrier of S2 holds

for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n))

let s be State of A; :: thesis: for s1 being State of A1 st s1 = s | the carrier of S1 holds

for s2 being State of A2 st s2 = s | the carrier of S2 holds

for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n))

let s1 be State of A1; :: thesis: ( s1 = s | the carrier of S1 implies for s2 being State of A2 st s2 = s | the carrier of S2 holds

for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n)) )

assume A6: s1 = s | the carrier of S1 ; :: thesis: for s2 being State of A2 st s2 = s | the carrier of S2 holds

for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n))

let s2 be State of A2; :: thesis: ( s2 = s | the carrier of S2 implies for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n)) )

assume A7: s2 = s | the carrier of S2 ; :: thesis: for n being natural Number holds Following (s,n) = (Following (s1,n)) +* (Following (s2,n))

let n be natural Number ; :: thesis: Following (s,n) = (Following (s1,n)) +* (Following (s2,n))

A8: (Following (s,n)) | the carrier of S1 = Following (s1,n) by A1, A3, A4, A5, A6, Th13;

A9: ( dom (Following (s,n)) = the carrier of S & the carrier of S = the carrier of S1 \/ the carrier of S2 ) by A3, CIRCCOMB:def 2, CIRCUIT1:3;

S1 tolerates S2 by A4, CIRCCOMB:def 3;

then A10: S1 +* S2 = S2 +* S1 by CIRCCOMB:5;

A1 +* A2 = A2 +* A1 by A4, CIRCCOMB:22;

then (Following (s,n)) | the carrier of S2 = Following (s2,n) by A2, A3, A4, A5, A7, A10, Th13, CIRCCOMB:19;

hence Following (s,n) = (Following (s1,n)) +* (Following (s2,n)) by A8, A9, FUNCT_4:70; :: thesis: verum